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If (x + 2)(x – 3) = 0 and x > 1/2, what is the value of x^-2

If $(x+2)(x-3)=0$ and $x>1/2$, what is the value of $x^{-2}$?

$-1/4$

$-1/9$

$0$

$1/9$

$1/4$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 18 of Section 6 of Practice Test 1. Those questions testing our knowledge of Exponents and Roots can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

We see that we want to evaluate $x^{-2}$, so the presence of an exponent suggests that we’ll use our Exponents and Roots math skill. We also have the product of two algebraic expressions is equal to $0$, which we recognize from our Solving Quadratic Equations math skill. Let’s keep what we’ve learned about these skills at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

We have the equation: $(x+2)(x-3)=0$

We have the inequality: $x>1/2$

We want to find the value of $x^{-2}$

Develop a Plan

Well, the question appears to be fairly straightforward. Still, let’s make a list of the plan we’ll use to find our solution.

We have an equation with $x$ in it, so let’s solve it for $x$

Make sure that our value for $x$ satisfies our inequality

Calculate the value of $x^{-2}$

Solve the Question

We remember from our Solving Quadratic Equations math skill a special case for when the product of two algebraic expressions is equal to $0$. Specifically, we know that for the product of two algebraic expressions to be equal to $0$, one of the algebraic expressions MUST be equal to $0$.

So from our equation, either $(x+2)=0$, in which case $x=-2$, OR $(x-3)=0$, in which case $x=3$. So from our equation, $x$ can have the values of $-2$ or $3$.

However, we do have an inequality for $x$, stating that $x$ must be greater than $1/2$. So we can discard the $x=-2$ solution, giving us $x=3$ as the only solution to the equation and inequality.

Excellent! Now all we have to do is solve $x^{-2}$, which we now know is $3^{-2}$. We see that we have a negative exponent, so let’s review how we simplify negative exponents, if we’ve forgotten how to do so, with the Negative Exponent Rule.

Concept Refresher – Negative Exponent Rule

We can convert negative exponents into positive exponents by just moving the entire base and exponent from the numerator to the denominator, or vice versa. We can also think of this as taking the reciprocal of a number (where we divide $1$ by that number) and then change the exponent to positive. For example:

$$5^{-7} = (1/5)^7$$

OR

$$(1/4)^{-5} = 4^5$$

Now let’s get back to the question at hand.

Applying the Negative Exponent Rule to simplify $3^{-2}$, we get:

$$3^{-2}=1/3^2=1/9$$

The correct answer is D, $1/9$.

What Did We Learn

Combining completely different math skills is a favorite way for the GRE to increase the difficulty of some of its math questions. However, we were wise to realize that we could use our math skills sequentially, by FIRST solving for $x$ and then calculating the value of $x^{-2}$. Divide and conquer is the way to go!